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Question 17 - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at Aug. 14, 2023 by

The set of all points where the function f(x)=x+|x| is differentiable, is

(a) (0,∞)                                    (b) (-∞,0)

(c) (-∞,0)∪(0,∞)                        (d) (-∞,∞)

This question is similar to Question-15 NCERT-Exemplar-MCQs

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Transcript

๐‘“(๐‘ฅ) = ๐‘ฅ+|๐‘ฅ| = {โ–ˆ(๐‘ฅ+๐‘ฅ, ๐‘ฅโ‰ฅ0@๐‘ฅโˆ’๐‘ฅ, ๐‘ฅ <0)โ”ค = {โ–ˆ(2๐‘ฅ, ๐‘ฅโ‰ฅ0@0, ๐‘ฅ<0)โ”ค Now, f(x) is a differentiable at x = 0 if LHD = RHD (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™) โˆ’ ๐’‡(๐’™ โˆ’ ๐’‰))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0) โˆ’ ๐‘“(0 โˆ’ โ„Ž))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐ŸŽ) โˆ’ ๐’‡(โˆ’๐’‰))/๐’‰ =(๐‘™๐‘–๐‘š)โ”ฌ(โ„Žโ†’0) (2(0) โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = ๐ŸŽ (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™ + ๐’‰) โˆ’ ๐’‡(๐’™ ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0 + โ„Ž) โˆ’ ๐‘“(0))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’‰) โˆ’ ๐’‡(๐ŸŽ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2โ„Ž โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 2โ„Ž/โ„Ž = ๐Ÿ Hence, we can say that f(x) is differentiable on R โˆ’ {๐ŸŽ} or (โˆ’โˆž,๐ŸŽ)โˆช(๐ŸŽ,โˆž) So, the correct answer is (c)

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